{"paper":{"title":"The Last Digit of $\\binom{2n}{n}$ and $\\sum\\binom{n}{i}\\binom{2n-2i}{n-i}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Walter Shur","submitted_at":"2013-02-01T14:03:26Z","abstract_excerpt":"Let $f_{n}=\\sum_{i=0}^n \\binom{n}{i}\\binom{2n-2i}{n-i}$, $g_{n}= \\sum_{i=1}^n \\binom{n}{i}\\binom{2n-2i}{n-i}$. Let $\\{a_k\\}_{k=1}$ be the set of all positive integers n, in increasing order, for which $\\binom{2n}{n}$ is not divisible by 5, and let $\\{b_k\\}_{k=1}$ be the set of all positive integers n, in increasing order, for which $g_n$ is not divisible by 5. This note finds simple formulas for $a_k$, $b_k$, $\\binom{2n}{n}\\ mod\\ 10$, $ f_{n}\\ mod\\ 10$, and $ g_{n}\\ mod\\ 10$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.0188","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}